具有凸-凹-凸非线性项的边值问题正解的确切个数
Exact Multiplicity of Positive Solutions for Convex-Concave-Convex Nonlinearities
摘要: 本文研究了具有凸-凹-凸非线性项的狄利克雷边值问题正解的分支曲线。通过时间映射分析法，证明了在非线性项为渐近次线性时，边值问题的正解分支曲线为S-型曲线，从而确定了边值问题的正解的确切个数。
Abstract: In this paper we study the bifurcation curve of the Dirichlet boundary value problem with convex- concave-convex nonlinearities. By using Time-map techniques, we prove that the bifurcation curve of the boundary value problem is S-shaped when the nonlinearities are asymptotic sublinear. Consequently, the exact multiplicity of positive solutions is determined.
文章引用：黄子饶, 白定勇. 具有凸-凹-凸非线性项的边值问题正解的确切个数[J]. 应用数学进展, 2017, 6(3): 317-326. https://doi.org/10.12677/AAM.2017.63037

参考文献

[1] Laetsch, T. (1970) The Number of Solution of a Nonlinear Two Point Boundary Value Problem. Indiana University Mathematics Journal, 20, 1-13. [Google Scholar] [CrossRef] 
[2] Wang, S.-H. and Long, D.-M. (2001) An Exact Multiplicity Theorem Involving Concave-Convex Nonlinearities and Its Application to Stationary Solutions of a Singular Diffusion Problem. Nonlinear Analysis, 44, 469-486. [Google Scholar] [CrossRef] 
[3] Hung, K.-C. and Wang, S.-H. (2011) A Theorem on S-Shaped Bifurcation Curve for a Positone Problem with Convex-Concave Nonlinearity and its Applications to the Perturbed Gelfand Problem. Journal of Differential Equations, 251, 223-237. [Google Scholar] [CrossRef] 
[4] Korman, P. and Li, Y. (2014) Exact Multiplicity of Positive Solutions for Concave-Convex and Convex-Concave Nonlinearities. Journal of Differential Equations, 257, 3730-3737. [Google Scholar] [CrossRef] 
[5] Brown, K.J., Ibrahim, M.M.A. and Shivaji, R. (1981) S-Shaped Bifurcation Curves. Nonlinear Analysis, 5, 475-486. [Google Scholar] [CrossRef] 
[6] Korman, P. and Li, Y. (1999) On the Exactness of an S-Shaped Bifurcation Curve. Proceedings of the American Mathematical Society, 127, 1011-1020. [Google Scholar] [CrossRef] 
[7] Huang, S.-Y. and Wang, S.-H. (2015) On S-Shaped Bifurcation Curves for a Two-Point Boundary Value Problem Arising in Theory of Thermal Explosion. Discrete and Continuous Dynamical Systems, 35, 4839-4858. [Google Scholar] [CrossRef] 
[8] Huang, S.-Y. and Wang, S.-H. (2016) Proof of a Conjecture for the One-Dimensional Perturbed Gelf and Problem from Combustion Theory. Archive for Rational Mechanics and Analysis, 222, 769-825. [Google Scholar] [CrossRef] 
[9] Wang, S.-H. (1994) On S-Shaped Bifurcation Curves. Nonlinear Analysis Theory Methods & Applications, 22, 1475-1485. ttps://doi.org/10.1016/0362-546X(94)90183-X
[10] Du, Y. (2005) Bifurcation and Related Topics in Elliptic Problems. Handbook of Differential Equations Stationary Partial Differential Equations, 2, 127-209. [Google Scholar] [CrossRef] 
[11] Y.Du, Y.Lou (2001) Proof of a Conjecture for the Perturbed Gelfand Equation from Combustion Theory, Journal of Differential Equations, 173, 213-230. [Google Scholar] [CrossRef] 
[12] Korman, P. (2006) Global Solution Branches and Exact Multiplicity of Solutions for Two Point Boundary Value Problems. Handbook of Differential Equations Ordinary Differential Equations, 3, 547-606. [Google Scholar] [CrossRef] 
[13] Korman, P., Li, Y. and Ouyang, T. (2005) Computing the Location and the Direction of Bifurcation. Mathematical Research Letters, 12, 933-944. [Google Scholar] [CrossRef] 
[14] Smoller, J. and Wasserman, A. (1981) Global Bifurcation of Steady-State Solutions. Journal of Differential Equations, 39, 269-290. [Google Scholar] [CrossRef] 

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